A light beam described by an LG mode carries well-defined OAM in the \(z\) -direction, which is related to the azimuthal phase factor \(e^{i\ell \phi }\) , where integer \(\ell \in (-\infty ,\infty )\) , called the topological charge, defines the OAM per photon in the beam [1]}. The other index characterizing the mode, \(p\in [0,\infty )\) , is called the radial index.
It is related to the number of dark rings in the intensity profile of \(u_{\ell p}\) , but does not have a straightforward connection with a physical quantity as is the case for \(\ell \) .
In recent years, however, the radial index has been the subject of theoretical works [2]}, [3]} that have enlightened its significance.
